We tend to think that our modern electronic devices are very energy-efficient so mechanical mainsprings etc. must be enough but they're not. After all, the (Intel i7) microprocessors have over 1 billion transistors per chip and each transistor has to consume some nonzero (and not "totally" negligible) energy, after all, to do an operation and they do billions of operations per second.
The mainsprings are an extremely lousy storage of energy. The total mechanical energy of such a mainspring may be calculated as $E=\int F\cdot d\ell$. This integral is comparable to the longitudinal force of the mainspring multiplied by the change of its length after it unwinds.
When the numbers are substituted, the mainspring 2824-2 in a wristwatch only contains 0.3 joules. One may basically view the energy as extensive and fill a volume with mainsprings. The energy stored in these mechanical devices will be 1530 joules per liter. It is just 1.5 kilojoules per liter.
This is tiny compared to the energy stored in gasoline – 35 megajoules per liter which is 20,000 times more concentrated energy than the energy in mainsprings. The energy stored in the same volume of batteries will be comparable to gasoline (because both of them are based on the chemical energy of electronic orbitals), just a little bit lower. Lithium-ion batteries have about 4 megajoules per liter, about 9 times lower than gasoline, see the table here
Lithium-ion batteries still store 3,000 times more energy than the mainspring of the same volume. It shouldn't be surprising microscopically. The mainspring only rearranges macroscopic pieces of the metal which only uses relatively small forces we may afford not to make the spring too dangerous when it cracks. On the other hand, the chemical energy (gasoline, lithium-ion) uses the much larger forces that keep the atoms together or apart etc. They store the near-maximum chemical energy in every atom or every pair of adjacent atoms, so to say.
The nuclear energy stored per kilogram is over 1 million times greater than the gasoline; and it is almost 1 billion times more concentrated energy than gasoline on the per-volume basis (because the uranium etc. is denser). An even more shocking comparison is the uranium-vs-mainspring comparison on the per-volume basis: uranium (nuclear) is about 1 trillion times more concentrated form of useful energy than the mainsprings. (If we could get lots of antimatter and annihilate it against matter, we would gain another factor of nearly 1,000 in the energy content per liter – almost 1 quadrillion times denser useful energy than in mainsprings – and no further improvement would be possible.)
For those reasons, the mainsprings are a romantic old-fashioned solution but it is not practical for the modern devices which consume much more energy than what a mainspring may give.
Ordinary people know quite something about how much energy batteries can carry. They have batteries e.g. in portable small vacuum cleaners (not to mention Tesla cars) and they may easily see that the amount of mechanical work that the vacuum cleaner (let alone car) may perform with new batteries would be enough to wind a mainspring many times.
Because this question is literally about the approximate calculation of energy, qualitative physical mechanisms behind different types of storages, and the same order-of-magnitude estimates (in atomic physics etc.) that a part of our physics PhD qualifying exams at Rutgers were full of, I strongly disagree with the users who classified this question as an "off-topic question on engineering" before they closed it.